1 Preferences. Completeness: x, y X, either x y or y x. Transitivity: x, y, z X, if x y and y z, then x z.

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1 1 Preferences We start with a consumption set X and model people s tastes with preference relation. For most of the class, we will assume that X = R L +. The preference relation may or may not satisfy the following properties. Completeness: x, y X, either x y or y x. Transitivity: x, y, z X, if x y and y z, then x z. Reasons for violation include framing effects, social choice, just-perceptible differences Monotonicity More of everything is better Strong monotonicity More of something is better Local non-satiation Continuity Convexity: y x, z x αy + (1 α)z x Strict convexity: y x, z x αy + (1 α)z x We also define x y x y but not y x. : x y x y and y x. If satisfies completeness and transitivity, we say that it is rational. Show that if is rational, then is transitive and irreflexive (there is no x such that x x). 1

2 is transitive, reflexive (x x), and symmetric (x y means y x) This shows that our explicit assumptions imply some other, implicit ones. For another exercise, show that is is strongly monotone, then it is monotone. Show also that if is monotone, then it is locally nonsatiated. Thus, strong monotonicity implies monotonicity implies local nonsatiation. Strong monotonicity is the strongest assumption; monotonicity is weaker one; and local nonsatiation is the weakest one. (E.g., tall fat man implies tall man implies man. Tall fat man puts the most restrictions on our individual and is therefore the strongest assumption.) 1.1 Examples of Preferences Leontief, perfect substitutes, log, etc. 1.2 Utility representation Ultimately, we want to model our preferences with utility function. Definition 1. u : X R represents if x y u(x) u(y). The following proposition shows that not all preferences can be modeled: Proposition 1. If has a utility representation, then it is rational. Question: do rational preferences always have a utility representation? No! Lexicographic preferences are a counterexample. Theorem 1. If is rational and continuous, then it is represented by a continuous utility function. There is an easy proof for the case where is also monotone. Exercise: show that lexicographic preferences are not continuous. 2

3 2 Revealed Preference Economists model behavior as constrained optimization. Question: what restrictions does this put on choice? Typically, we observe (chosen) consumption bundles together with (vectors of) prices. Let s say we are given observations (x 1, p 1 ),..., (x T, p T ). Definition 2. u rationalizes these observations if u(x t ) u(x) for every x R L + such that p t x t p t x. If the preferences (x 1, p 1 ),..., (x T, p T ) are rationalized by a locally nonsatiated utility function, then the following must hold: 1. The consumer s income at time t is p t x t. 2. u(x t ) u(x) for every x R L + such that p t x t p t x. 3. u(x t ) > u(x) for every x R L + such that p t x t > p t x. (2) and (3) imply that if p t x s p t x t then p s x t p s x s. The property in red is called the Generalized Weak Axiom of Revealed Preference (GWARP). Choices must satisfy it in order to be rationalizable. Is GWARP sufficient for rationalizability? No! Write x t Rx if p t x t p t x and x t Rx if p t x t > p t x. We can now restate GWARP as x t Rx s not x s P x t. Utility representation requires that for any subset of observations {t 1, t 2,..., t n }, x t1 Rx t2, x t2 Rx t3,..., x tn 1 Rx tn not x tn P x t1. This is called the Generalized Axiom of Revealed Preference (GARP). These axioms are generalizations of the following: WARP: x t Rx s and x t x s not x s Rx t. 3

4 SARP: x t1 Rx t2, x t2 Rx t3,..., x tn 1 Rx tn and x t1 x tn not x tn Rx t1. For an exercise, show that WARP implies GWARP and SARP implies GARP. 2.1 Andreoni and Miller Basic idea: individuals exhibit social preferences. Can these preferences be explained by a reasonable utility function? Experimental design: π s +p π o = m. Money allocated to self and other with different budget sets. Divide 60 tokens, hold X at 1 point each, pass Y at 2 points each. Points converted to dollars. Several treatments (see Table 1). 176 subjects, 18 violated one more of the revealed preference axioms. 43% of subjects behaviors could be explained by one of the following three preferences: Leontief (50,50), selfish, perfect substitutes. 3 Demand Theory The utility maximization problem (UMP) is given by s.t. max x u(x) px w x 0 The problem above has a solution if p >> 0 and u is continuous. We denote the solution by x(p, w). It s called the Walrasian Demand Correspondence. Walrasian demand has the following properties: x(αp, αw) = x(p, w) for all α > 0 4

5 px = w for all x x(p, w) If u represents preferences that are convex, then x(p, w) is a convex set. If there preferences are strictly convex, then x(p, w) is a singleton. The UMP can be characterized by its first order conditions (FOCs). 3.1 Kuhn-Tucker Conditions Consider the problem max f(x) x R N s.t. g 1 (x) = b 1... g M (x) = b M h 1 (x) c 1... h K (x) c M Theorem: if x is a local maximum satisfying the constrains then there are multipliers λ m R, one for each equality constraint, and λ k R, one for each inequality constraint, such that: 1. f(x ) x n = M λ m g m(x ) x n + K λ k h k (x ) x n for all n 2. λ k (h k (x ) c k ) = 0 for all k In words, the second condition says that λ k = 0 for each inequality constraint that does not bind. Sometimes, the conditions above are stayed in terms of a Lagrangian function. Define L(x, λ) = f(x) M λ m ( b m g m (x)) K λ K ( c k h k (x)). Then, (1) is equivalent to L(x,λ) x n = 0 for all n. 5

6 It is common for one of the constraints to take the form of x l 0. In this case, the FOC for x l changes to L(x,λ) x n 0 (show this). Example: maximize f(x, y) = xy subject to x + y 2 2 and x, y FOCs for UMP If u is continuously differentiable, then u(x ) x l λp l (= if x l > 0). If the solution is interior, then u l (x ) u k (x ) = p l p k. This is the familiar condition that the ratio of prices is equal to the Marginal Rate of Substitution (MRS). In the UMP, λ has a nice interpretation: this is the marginal utility obtained from relaxing the budget constraint a little bit. For an exercise, show that u(x ) w = λ. Another exercise: Solve the utility maximization problem for u(x 1, x 2 ) = x α 1 x 1 α Indirect Utility We will find it useful to work with the indirect utility function given by v(p, w) = u(x(p, w)). Suppose preferences are l.n.s. and u is continuous. Then the indirect utility function has the following properties: Homogeneity of degree 0 Strictly increasing in w and non-increasing in p l for any l Continuous in p and w Properties (1) and (2) are easy to verify. Do it! 6

7 3.4 Expenditure Minimization The expenditure minimization problem (EMP) is given by min x 0 px s.t. u(x) u Theorem: Suppose u is a continuous function representing l.n.s. preference on R L + and p >> 0. Then 1. If x is optimal for UMP, then it is optimal for EMP with u = v(x ). Moreover, px = w. 2. If x is optimal for EMP, then it is optimal for UMP with w = x p. Moreover, the maximized utility is given by u. The solution to the EMP, denoted as h(p, u) is the Hicksian Demand Correspondence. The function e(p, u) = ph(p, u) is called the Expenditure Function. These objects have the following properties: h is homogenous of degree 0 in prices e is homogenous of degree 1 in prices e is a concave function of prices If e id differentiable at (p, u) and h is single valued then D p e(p, u) = h(p, u). 3.5 Slutsky It can be shown that h l (p, u) p k = x l(p, w) p k + x l(p, w) w x l(p, w) or equivalently x l (p, w) p k = h l(p, u) p k x l(p, w) w x l(p, w) 7

8 4 Uncertainty So far, we have considered a world without uncertainty. Now, let X = {x 1,..., x N } denote a set if uncertain outcomes. A lottery (p 1,..., p N ) with p n 0 for all n and n p n = 1 is an assignment of probabilities to outcomes. Given simple lotteries (p 1,..., p K ), a compound lottery (L 1,..., L K ; α 1,..., α K ) with α k 0 for all k and k α k = 1 yields lottery L k with probability α k. We are interested utility representations for preferences over the space of lotteries L. The preference relation on the space of lotteries is continuous if L, L, L L, the sets {α [0, 1] : αl + (1 α)l L } and {α [0, 1] : αl + (1 α)l L } are closed. Note: a set A is closed if and only if for every sequence x n x with x n A for every n, x A. Proposition: If is continuous, there exists a utility representation for. 4.1 Independence satisfies independence if for all L, L, L L and α (0, 1) we have L L αl + (1 α)l αl + (1 α)l Exercise: show that if satisfy independence, then for all L, L, L L and α (0, 1), L L αl + (1 α)l αl + (1 α)l 8

9 4.2 Expected Utility A utility function U : L R has an expected utility form if there is an assignment of numbers {u 1,..., u N } such that every simple lottery U(L) = p 1 u p N u N. A utility function U with an expected utility form is called a von Neumann-Morgenstern (vnm) expected utility function. Proposition: U : L R has an expected utility form if and only if it is linear. Proposition: Suppose U : L R is a vnm utility representing. Then Ũ : L R is another vnm utility representing if and only if β > 0 and γ such that Ũ = βu + γ. Exercise: Show that if α is represented by a vnm utility function, then is continuous and satisfies the independence axiom. Theorem (Expected Utility Theorem): If satisfies continuity and independence, then it is represented by a vnm utility function. 4.3 Risk Aversion We now denote money by a random variable x with F ( x) = P rob(x x). This generalizes the analysis so far. E.g., consider a lottery with three outcomes P (x = 1) = 1, P (x = 4) = 1, P (x = 6) = 1. Then, if x < 1 1 if x [0, 4) 4 F (x) = 3 if x [4, 6) 4 1 if x 6 Expected utility is defined as U(F ) = u(x)df (x), where u is the Bernoulli utility, as before. We can now define risk aversion. An individual is risk averse if E(u(x)) u(e(x)). 9

10 An individual is strictly risk averse if E(u(x)) < u(e(x)) for all non-degenerate F. An individual is risk neutral if E(u(x)) = u(e(x)). Risk aversion is equivalent to concavity of the Bernoulli utility function u( ). This is intuitive. Decreasing marginal utility a gain of 1 is worse than a loss of 1, so a bet that gives a gain of 1 and a loss of 1 with equal probability is not worth taking. Commonly used u(x) s include: Linear: u(x) = x Constant absolute risk aversion (CARA): u(x) = e αx for α > 0 Quadratic: u(x) = (α x) 2 for x < α Constant relative risk aversion (CRRA): u(x) = x1 γ 1 γ for γ > 0 One useful concept for studying risk aversion is the certainty equivalent, denoted c(f, u). This is amount of money that gives the agent the same utility as the gamble F : u(c(f, u)) = u(x)df (x) all F. For an exercise, show that if an agent is risk averse, then c(f, u) xdf (x) for Another useful concept is the probability premium. If a agent is offered a gamble that gives ɛ and ɛ > 0 with some probabilities, the probability premium π(w, ɛ, u) is the amount by which the probability of winning must be larger for the agent to be indifferent between the gamble and a certain outcome w: u(w) = ( π(w, ɛ, u)) u(w + ɛ) + ( 1 2 π(w, ɛ, u)) u(w ɛ) w, ɛ. For an exercise, show that if the agent is risk averse then π(w, ɛ, u) 0 for all 10

11 4.4 Arrow-Pratt We can use the following measures to compare risk aversion of different utility functions: The Arrow-Pratt measure of absolute risk aversion A(x) = u (x) u(x) The Arrow-Pratt measure of relative risk aversion R(x) = u (x) u(x) x Exercise: Let u 1 and u 2 be two utility functions. Show that A 1 (x) A 2 (x) for every value of x if and only if there exists and increasing concave function ψ such that u 2 (x) = ψ(u 1 (x)) at all x. I.e., u 2 is more concave than u Insurance We can use the concepts above to study insurance. Let an agent start with an asset level w. There is a probability π of a loss D. The agent can purchase insurance at price q. If the agent buys α units of insurance, then his payoff is w αq with probability 1 π w αq D + α with probability π This means his expected wealth is w πd + α(π q). Imagine that the price of insurance is acturially fair, which means q = π. Then the expected wealth of the agent is w πd no matter how much insurance he buys. He therefore sets α = D to reach this amount with certainty. I.e., the agent insures himself fully. 11

12 4.6 Asset Selection Consider the problem of buying a portfolio of assets (α 1,..., α N ) with returns (z 1,..., z N ). The utility of holding the portfolio of assets is U(α 1,..., α N ) = u(α 1 z α N z N )df (z 1,..., z N ). If these assets have prices (p 1,..., p N ), we can use what we learned about utility maximization to study demand of these assets, use revealed preference theory to see if such a utility function can explain the observations, etc. 12

13 5 Paradoxes We consider two paradoxes to illustrate some difficulties in applying expected utility theorem 5.1 Newcomb s Paradox This paradox shows that sometimes states should be defined as functions from actions to outcomes. I.e., assuming that your choice has no effect on the state can be a mistake. It starts with the following problem: There are two boxes. A clear box, which has 1000 dollars, and an opaque box, which may have dollars and may have nothing. You are asked to choose between both boxes and the opaque box only. There is a caveat: you observed 1000 people choose before you. 500 choose both boxes and found the opaque box empty. 500 chose the opaque box and they are now millionaires. Expected utility predicts that you should choose both boxes, since regardless of the state (there is money in the opaque box, there is no money in the opaque box), you are better off. This ignores the apparent relationship between states and actions. Construct a state space as follows: {(1, 1), (1, 0), (0, 1), (0, 0)}. The numbers are 1=the money is there, 0=the money is not. The first number is what happens if you re greedy, and the second number is what happens if you re modest. Now, there is no apparent dominance. (1,1) (1,0) (0,1) (0,0) Greedy Modest In some states it s better to be greedy, and in some states better to be modest. 13

14 5.2 Monty Hall This paradox shows that the way in which information is attained sometimes matters. In this case, the state space should reflect how information is attained. In this problem, you are at a game show. There are three doors: A, B, and C. Two doors contain a goat and one door contains a car. The probability of the prize being behind each of the doors is 1. You get to choose a door, then the host opens 3 one of the doors with a goat. Assume you choose door A and the host opened door B (which contains a goat). Do you change your decision or not? The answer is: yes, you should change your decision. This is because the host s decision contains information. Intuitively, modify the problem in the following way. There are 100 doors, one with a prize. You choose a door. Then the host opens 98 doors with goats. Do you change your mind now or not? The problem with the usual way of thinking about the problem is that you think of the state space as where the prize is. To think about it correctly, the state is (a) where the prize is and (b) what door the host opened. Assume that you chose door A. Then the probability of the prize being behind each of the doors for every state can be illustrated in the following matrix: OA OB OC A B C The 0 s on the diagonal reflect that the host doesn t open the door with a prize. The 0 s in the first column reflect that the door will not open the door you chose (door A). Since the unconditional probability of the prize being behind each of the doors is 1, this leaves us no choice in the bottom two rows. Conditional on the prize 3 beind in A, it is equally likely that the host opens door B and door C. It is easy to calculate that P (A OB) = 1 and P (C OB) = 2. Thus, if the host opens door B, you 3 3 should switch. 14

15 6 Prospect Theory Preference for Certainty Consider the following pair of choices. Problem 1 : Choose between (A) 2500 with probability 0.33, 2400 with probability 0.66, 0 with probability 0.01 (B) 2400 with certainty. Problem 2 : Choose between (C) 2500 with probability 0.33, 0 with probability 0.67 (D) 2400 with probability 0.34, 0 with probability 0.66 Typically, subjects choose B in Problem 1 and C in Problem 2. This is called Allais paradox because it is a violation of expected utility. Here is another problem: Problem 1 : Choose between (A) 50% chance of winning a 3 week tour of England, France, and Italy (B) A one week tour of England with certainty. Problem 2 : Choose between (C) 5% chance of winning a 3 week tour of England, France, and Italy (D) 10% chance of winning a one week tour of England And another: Problem 1 : Choose between (A) 45% chance of winning 6000, and 65% chance of winning nothing (B) 90% chance of winning 3000, and 10% chance of winning nothing Problem 2 : Choose between (C) 0.1% chance of winning 6000 and 99.9% chance of winning nothing (D) 0.2% chance of winning 3000 and 99.8% chance of winning nothing 15

16 In the latter two problems, people usually chose B in the first choice and C in the second, which also violated expected utility. Kahneman and Tversky (1979) considered all three of these violations to be a generalization of the same phenomenon: If (y, pq) is equivalent to (x, p), then (y, pqr) is preferred to (x, pr). The Reflection Effect Kahneman and Tversky (1979) observed that risk aversion in the domain of gains is accompanied by risk seeking in the domain of losses. For example, the majority of subjects accept a risk of 80% of losing 4000 in preference to a sure loss of 3000 although the gamble has lower expected value. Moreover, just like in the positive domain, outcomes with are obtain with certainty are overweighed relative to uncertain outcomes. For example, consider the following pair of problems. Problem 1 : Choose between (A) 45% chance of losing 6000, and 65% chance of losing nothing (B) 90% chance of losing 3000, and 10% chance of losing nothing Problem 2 : Choose between (C) 0.1% chance of losing 6000 and 99.9% chance of losing nothing (D) 0.2% chance of losing 3000 and 99.8% chance of losing nothing Subjects typically choose A in problem 1 and D in problem 2. Note that this violates expected utility! This violation can be explained by the following principle: If ( y, pq) is equivalent to ( x, p), then ( y, pqr) is worse than ( x, pr). Gain-Loss Asymmetry This is the final piece of the puzzle. Consider the following pair of problems. Problem 1 : In addition to what you own, you have been given 1000 dollars. between Choose 16

17 (A) 50% chance of winning 1000, and 50% chance of winning nothing (B) 500 with certainty. Problem 2 : In addition to what you own, you have been given Choose between (C) 50% chance of losing 1000 and 50% chance of losing nothing (D) -500 with certainty. People typically choose B in the first problem and C in second. But the problems are the same! Evidently, subjects didn t take the bonus into account when making their decisions. This led Kahneman and Tversky to conclude that the relevant variables of interest are changes in wealth rather than absolute wealth levels (which is what matters in the expected utility world). Theory Kahneman and Tversky hypothesized that utility from a regular gamble (x, p; y, q) is evaluated in the following way: U(x, p; y, q) = π(p)v(x) + π(q)v(y). A regular gamble is a gamble where either p + q < 1 or x 0 y or y 0 x. v is called the value function and π is called the decision weight. Kahneman and Tversky emphasize that a decision weight is different from your belief about the event because in general it does not satisfy the probability axioms. For strictly positive and strictly negative gambles (p + q = 1 and x > y > 0 or 0 > y > x), utility is evaluated differently. Subjects take the less extreme of these amounts and factor it out of the problem. Then, U(x, p; y, q) = v(y) + π(p)(v(x) v(y)) The thing to notice above is that the decision weight is only applied to the uncertain part of the gamble. Thus, your utility is what you get for sure plus (or minus) the difference in utility due to the uncertain component. 17

18 Properties of the value function Kahneman and Tversky use some experimental data to argue that: v is concave for gains but convex for losses v has a steeper slope in the loss domain than in the gain domain (loss aversion) To see the first argument, consider the following pair of problems. Problem 1 : Choose between (A) 25% chance of winning 6000, and 75% chance of winning nothing (B) 25% chance of winning 4000, 25% chance of winning 2000, and 50% chance of winning nothing Problem 2 : In addition to what you own, you have been given Choose between (C) 25% chance of losing 6000, and 75% chance of losing nothing (D) 25% chance of losing 4000, 25% chance of losing 2000, and 50% chance of losing nothing The typical pattern of choices is B and C. Applying Kahneman and Tversky s utility function for regular gambles, we get π(.25)v(6000) > π(.25)v(4000) + π(.25)v(2000) and π(.25)v( 6000) < π(.25)v( 4000) + π(.25)v( 2000) We can rewrite this as and v(6000) v(4000) < v(2000) v( 6000) v( 4000) > v( 2000) which is consistent with the hypothesis that v is concave for gains and convex for losses. To argue for loss aversion, Kahneman and Tversky note that if x > y 0, (y,.5; y,.5) is typically preferred to (x,.5; x,.5). 18

19 Therefore, v(y) + v( y) > v(x) + v( x) or v( y) v( x) > v(x) v(y) Letting y approach x, this implies that v ( x) > v(x). Properties of the decision weights Kahneman and Tversky argue that: π(p) > p for small probabilities (overweighting) π(p) + π(1 p) < 1 generally (subcertainty) or Recall that people prefer (6000, 0.1) to (3000, 0.2). Hence, π(0.1)v(6000) > π(0.2)v(3000) π(0.1) π(0.2) > v(3000) v(6000) > 1 2, where the last inequality follows from the concavity of v. Thus π(p) > p, at least for small probabilities. What Kahneman and Tversky call subcertainty follows from their specification of the utility function and the Allais paradox. 7 Cautious Expected Utility The discussion below follows a recent paper (Cerreia-Vioglio, Dillenberger, Ortoleva 2014) that provides an alternative to prospect theory. The paper considers lotteries over an interval [w, b] R. Let δ x denote the lottery that pays you x [w, b] for sure, and the space of all lotteries. This paper takes the certainty effect as a starting point and asks the question of what axiom can capture it. The axiom that the authors come up with is the following. 19

20 Negative Certainty Independence: For any p, q, x [w, b], and λ [0, 1], p δ x λp + (1 λ)q λδ x + (1 λ)q. In words, if p is better than a certain thing, introducing the same amount of additional uncertainty on both sides still makes p a better thing. Homework: Show that NCI implies Kahneman and Tversky s definition of the certainty effect: If (y, q) is equivalent to (x, 1), then (y, qr) is preferred to (x, r). Show that NCI implies convexity. CDO show that together with some simple basic axioms (rationality, continuity, monotonicity), we get what s called a Caution Expected Utility representation. Cautious Expected Utility: Let be a preference relation over, and let c(p, v) denote the certainty equivalent of p with Bernoulli utility v. We say that has a CEU representation if there exists a set W of strictly increasing and continuous utility functions on [w, b] such that p q V (p) V (q), where V (p) = inf v W c(p, v). In words, the agent here doesn t know what her utility function over [w, b]. She only knows it is contained in W. To compute her utility from a lottery p, she considers all possible (expected utility) valuations E p (v) and takes the lowest possible ones. It is in this sense that agent is cautious. Note that Cautious Expected Utility contains the usual kind of Expected Utility as a special case. This is true if the set W has one element. Note also that Prospect Theory contains expected utility as a special case, as well! This is true if π(p) = p. Thus, there as a point (and this point is Expected Utility) at which Prospect Theory and Cautious Expected Utility coincide. On the other hand, CEU is rich enough to also accommodate violations of Expected Utility such as the Allais Paradox. 20

21 Homework: Let [w, b] = R +, and W = {u 1, u 2 }, where u 1 (x) = exp( βx), β > 0; and u 2 (x) = x α, α (0, 1). Let α = 0.8 and β = and consider two problems. Problem 1: a choice between A = δ and B = 0.8δ δ 0. Problem 2: a choice between C = 0.25δ δ 0 and D = 0.2δ δ 0. Show if the agent has a Cautious Expected Utility representation with the set W, then A B and D C. The main result of CDO is the following. Theorem: Let be a preference relation on. The following statements are equivalent: 1. The relation satisfies rationality, continuity, weak monotonicity, and NCI. 2. There exists a CEU representation of. 21

22 8 Time Preferences Assume now that there are t = 1, 2,..., T periods, so that a consumption bundle is given by (x 1, x 2,..., x T ). Note that we can use what we learned above at preference relations to study the consumer s preferences and choices! Economists typically assume that preferences are additively separable over time, so that U(x 1,..., x T ) = T u t (x t ). t=0 We typically assume even more structure, i.e. that u t (x t ) = β t u(x t ), where β (0, 1) is the agent s discount factor. This captures the intuition that preferences don t change over time and that people are impatient. Note that E[U(x 1,..., x T )] = T β t E(u(x t )). t=0 Example: Let U(x 1, x 2 ) = log(x 1 )+βlog(x 2 ). Assume that the consumer receives income m 1 in period 1 and m 2 in period 2. Assume further that the consumer can borrow and lend at an interest rate r. Derive the consumer s consumption bundle (x 1, x 2 ) as a function of β, r, m 1, andm 2. Derive the conditions on β, r, m 1, andm 2 under which the consumer is a borrower, and provide an intuitive explanation. 8.1 Hyperbolic Discounting There is a lot of experimental evidence showing that a better way of modeling time preferences is: T U(x 1,..., x T ) = u(x 0 ) + β β t E(u(x t )). This is sometimes called the β δ model. Here, there is more discounting between today and tomorrow than between any other two adjacent periods. One type of experiment to show this asks people to choose between an apple now and two apples tomorrow, and also between an apple a year from now and two apples 22 t=1

23 a year from now and one day. People typically choose one apple now and two apples a year and a day from now. (Show that this is inconsistent with the exponential discounting model.) 8.2 Rubinstein, 2003 This paper shows that while it s easy to construct experiments that reject exponential discounting, one can also construct experiments that reject both exponential and hyperbolic discounting. Rubinstein considers a more general model of hyperbolic discounting, where U(x 0,..., x T ) = u(x 0 ) + T ( δ s )u(x t ). t=1 s=1,...,t Experiment 1 of Rubinstein, 2003 Q1 Imagine that you have to choose between the following two options: A) Receiving $ on June 17th B) Receiving $ on June 17th What will be your choice? Q2 Imagine that you have to choose between the following two options: A) Receiving $ on June 16th B) Receiving $ on June 17th What will be your choice? Homework: Show that Delay in Q1 and No delay in Q2 is not consistent with either exponential and hyperbolic discounting. 8.3 Risk Preferences Are Not Time Preferences Andreoni and Sprenger is an interesting paper that explores the possibility that risk and time preferences are intertwined. This is because what happens in the future is uncertain. Consider expected utility from a prospect that gives consumption c t at time t 23

24 with probability p 1 and 0 otherwise, and c t+k at time t + k with probability p 2 and 0 otherwise. Then, U = p 1 δ t u(c t ) + p 2 δ t+k u(c t+k ) + ((1 p 1 )δ t + (1 p 2 )δ t+k )u(c 0 ). Suppose that utility is maximized subject to the constraint (1 + r)c t + c t+k = m. The first order conditions give u (c t ) δ k u (c t+k ) = (1 + r)p 2 p 1. Notice that the optimal consumption bundle depends on the ratio of probabilities and not what the actual probabilities are. I.e., intertemporal allocation depend only on relative risk. For example, if a sooner payment is made 80 percent of the time and a later payment 40 percent of the time, the optimal consumption bundle should be the same as if a sooner payment is made 40 percent of the time and a later payment 20 percent of the time Experiment Subjects choose between allocating tokens to a sooner payment (7 days from now) and a later payment (28, 56 days) by allocating tokens out of 100 between sooner and later payments. a t+k =20 cents per token, a t =from 0.14 to 0.20 cents per token. Note that this plays around with interest rates. (p 1, p 2 ) {(1, 1), (0.5, 0.5), (1, 0.8), (0.5, 0.4), (0.8, 1), (0.4, 0.5)}. In conditions 1 and 2, both expected utility theory and prospect theory predict the same choice In conditions 3 and 4, expected utility predicts the same choice but prospect theory does not In conditions 3 and 4, expected utility predicts the same choice but prospect theory does not 24

25 9 Strategic Decisions We start with a mathematical definition of what a game it is. A normal form game is ((A i ) i=1,...,n, (u i ) i=1,...,n ), where i A i A = n i=1 Ai, and u i : A R i. is an action set, Sexes. I = {1,..., n} is the set of players. Assume A i is finite for all i. Examples: Coordination, Matching pennies, Prisoner s dilemma, Battle of the Let S i = (A i ) = {(s(a i 1),..., s(a i k i )) : i, s(a i ) 0, A i s(a i ) = 1}. A mixed extension of a normal form game is ((S i ) i=1,...,n, (u i ) i=1,...,n ), where S i = (A i ), S = n i=1 Si and u i : S R is defined by u i (s 1,..., s n ) = a A u i (a) We write P r s (a) = n i=1 si (a i ) A. n s i (a i ). We will now study three important solution concepts in game theory: dominance, rationalizability, and probably the most famous one of all, Nash Equilibrium. i=1 9.1 Mixed vs. Correlated Strategies Imagine there are two other players, player 1 with actions (L,R) and player 2 with actions (T,B). What are the beliefs we can have about the players actions? If we take a as a state, then the possible states from our perspective are (L,T),(L,B),(R,T),(R,B). Then the set of possible beliefs is the set of all possible probability distributions over these states. We denote this set by (A) and let µ (A) denote generic elements. Another possibility is to recognize that the probabilities of actions are induced by mixed strategies s = (s 1, s 2 ). Then {(P r s (a)) a A : s S} 25

26 is the set of probabilities on A induced my the mixed strategies. Notice that {(P r s (a)) a A : s S} is NOT equal to (A). Here is a belief that cannot be generated by mixed strategies: L R T B This belief is reasonably though if players are allowed to correlate their behavior. I.e., they both flip a SINGLE coin. If it s heads they choose (T,L) and if it s heads they chose (B,R). This kind of strategy is called a correlated strategy. 9.2 Dominance Example: Show that in the game below, the player can get a better payoff by mixing T and M than by playing B, no matter what his belief is about what his partner is doing. L R T 3 0 M 0 3 B 1 1 We say s i S i strictly dominates a i A i iff for all a i u i (s i, a i ) > u i (a i, a i ). Alternatively, or s i D 2 a i s i S i u i (s i, s i ) > u i (a i, s i ) s i D 3 a i µ (A i ) u i (s i, µ) > u i (a i, µ) Exercise: s i D 3 a i s i D 2 a i s i D 1 a i. Example: Note that T and L are both dominated in the game below. L R T -2,-2-10,-1 B -1,-10-5,-5 26

27 This leads to the counter-intuitive prediction of playing (B,R). Of course this doesn t happen in real life. 9.3 IESDA We illustrate this with examples: L R T 0,-2-10,-1 B -1,-10-5,-5 L R T 3,0 0,1 M 0,0 3,1 B 1,1 1,0 Related concept: IEWDA. Example: Using the game below, show that the set of actions surviving IEWDA is not unique. L R T 1,1 0,0 M 1,1 2,1 B 0,0 2,1 9.4 Rationalizability We start with an example: L R T 3 0 M 0 3 B x x If x < 3/2, B is never a best response. Dually, you can find a mix that strictly dominates B. At x = 3/2, B is a best response. Dually, you cannot find a mix that strictly 27

28 dominates B. This example suggests that an action is never a best response if and only if it is strictly dominated by a strategy. Definition: An action a i A i is never a best response if there is no µ (A i ) such that u i (a i, µ) u i (b i, µ) for all b i. Theorem: An action a i A i is strictly dominated if it is never a best response. (Show it!) In fact, this is an if and only if in 2 person games... but we skip the proof here. Thus, eliminating strategies that are never a best response eliminates at least as many strategies as the number of strategies that are strictly dominated. Moreover, as with IESDA, we can iterate the process of eliminated NBR strategies. Once I eliminate mine, you can take into account the remaining ones and eliminate yours, etc. The strategies that survive are called the rationalizable strategies. Example: find the rationalizable strategies of the game below. b 1 b 2 b 3 b 4 a 1 0,7 2,5 7,0 0,1 a 2 5,2 3,3 5,2 0,1 a 3 7,0 2,5 0,7 0,1 a 4 0,0 0,-2 0,0 10,-1 Exercise: show that the rationalizable strategies are a 1, a 2, a 3 for player 1 and b 1, b 2, b 3 for player 2. Note that for each of the rationalizable strategies a player can construct a chain of reasoning that justifies is. I play b 2 because I believe that player 1 will play a 2 which is reasonable if he thinks I play b 2 etc. 9.5 Nash Equilibrium We first define what it means to be a best response. We already talked about this informally, but here is the formal definition. 28

29 Definition (Pure Best Reply) P BR i (s) = {a i A i : b i A i u i (b i, P r s i)}. u i (a i, P r s i) Above, P r s i (A i ) with P r s i(a i ) = j i sj (a j ). Definition (Best Reply) BR i (s) = {s i S i : a i A i u i (s i, P r s i) u i (a i, P r s i)}. Example: find the BR correspondences of the game below. L R T 3,1 0,0 B 0,0 1,3 Proposition: For every s, BR i (s) is closed, convex, nonempty, and equal to the mixed strategies concentrated on P RB i (s). Definition: BR(s) = n i=1 BRi (s). Note that this is a correspondence from S to S. It is closed, convex, and nonempty. Definition: A Nash Equilibrium of a normal form game is a strategy profile ŝ such that ŝ BR(ŝ). That is, i Example: find NE of the game below. L R T 1,1 0,0 B 0,0 0,0 Example: find NE of the game below. L R T 3,1 0,0 B 0,0 1,3 ŝ i BR i (ŝ). 29